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Nomenclature
a [mm] center distance
b mm ace w t
ß [°] helix angle
c y mm•µmmean value of mesh stiffness per
unit face width
εα [-] transverse contact ratio
F t normal transverse load in plane
o act on
F n [N] load, normal to the line of contact
F
transverse tangential load at
reference cylinder per mesh
f a [-]component o equ va ent
misalignment
K A - app cat on actor
K α - transverse load factor
K ß [-] ace oa actor
K [-] dynamic factor
T 1 m torque at pinion
pC [N/mm2]
ertz an contact stress at t e
pitch point
- gear rat o
H 0 [N/mm2] nominal contact stress
H 1 [N/mm2] contact stress on p n on
v [m/s] tangential velocity
Z B/D [-]Single-pair tooth contact factor
for pinion wheel
Z E - e ast c ty actor
Z H [-] zone factor
Z β [-] helix angle factor Z ε - contact ratio factor
Influences of Load Distribution and
Tooth Flank Modifications as Considered in aNew, DIN/ISO-Compatible Calculation Method.
Bernd-Robert Höhn, Peter Oster and Gregor Steinberger
(This article first appeared in the Proceedings of
IDETC/CIE 2007 ASME 2007 International Design
Engineering Technical Conference & Computers and
Information in Engineering Conference, September
4–7, 2007 in Las Vegas.)
Introduction
Modern gears are designed with helical teeth to improve
their noise behavior and the pitting load capacity. Therefore,
a ver e ca cu at on met o to eterm ne t e p tt ng oa
capacity of helical gears is essential.
ISO 6336-2 and DIN 3990–Part 2 calculate a higher-
endurance torque for helical gears than for spur gears of the
ame size and material, but experimental research on the pittingoa capac ty o e ca spur gears s ac ng e s. , .
Management SummaryIn experimental analyses, the pitting load capacity of
case-carburized spur and helical gears is determined in
back-to-back test rigs.
Included for testing are one type of spur gear and eight
types of helical gears, with tests for the determination
of influences of varying load distribution, overlap ratio
and transmission ratio. The test results are presented
and evaluated on the basis of the pitting load capacity
calculation methods of ISO 6336-2/DIN 3990–Part 2.
A new DIN/ISO-compatible calculation method for
pitting load capacity is presented. This new calculation
method analyzes helical gears more adequately than ISO
6336-2/DIN 3990–Part 2, and has the ability to consider
tooth flank modifications. The new calculation method is
applied to test results and gears of a calculation study. It
shows better agreement with the experimental test results
than the present ISO 6336-2/DIN 3990–Part 2.
Pitting Load Capacityof Helical Gears
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GEAR TECHNOLOGY 00
47
Table 1: Gear geometry of the test gears
type Gk, S0, Slk and Stk
Test Type Gk S0 Slk Stk
center
distancea mm 112.5
normal
module mn mm 4. 4.2
number
of teeth z
1 /z
24/2 22/24
ace width b mm 27.6
helix angle β ° 0 29
addendum
odification
factor
x 1
- 0.270 .100
x 2
- 0.266 .077
tip diameter1
mm 119.4 117.7
2mm 123.9 127.3
working pitch
iameter
d w1
mm 110.2 107.6
d w
mm 114.8 117.4
normal
pressure
angle
α ° 20
ransverse
contact ratio ε
α1.5
verlap ratio ε 0 1.0
tooth ank modi cation
generated
relie- - no no no yes
tip relief
pinion/wheel
short/
short
long/
long
hort/
hort
0%
20%
40%
60%
80%
100%
120%
reference
dgvdfxchn
f g h f h f v
� = 0 °
� H
� = 29 °� = 21 °� = 14 °
F gure —Endurance contact stress of the spur and hel cal test gears
accord ng to ISO 6336-2/DIN 3990–Part 2 (Ref. 0).
hus the research project FVA 284 I/II “Helical Gear”
e . was n t ate to ana yze t e p tt ng oa capac ty
o e ca gears, t eoret ca y an exper menta y. pur an
helical gears of the same size and material batch were tested
to determine the influence of the helix angle. The endurance
contact stresses were calculated according to ISO 6336-2/
– art , ase on t e exper menta y eterm ne
at gue-en ura e torques. e resu ts are s own n gure
, which shows that the expected pitting load capacity ofσ
H 0 = 1, 00 N/mm is only reached by the spur gears, not
by the helical gears. Consequently, the calculation methods
accor ng to - – art eterm ne t e
p tt ng oa capac ty o spur gears more accurate y t an or
helical gears.
n order to confirm these results, additional tests (Ref.
2) were conducted. These tests prove the main result of
e . — .e., t e ca cu ate p tt ng oa
capac ty or e ca gears s too g . onsequent y, t e
calculation of helical gears according to ISO 6336-2 has to
be reconsidered.
Optimum tooth flank modifications for uniform pressure
str ut on resu t n max mum p tt ng oa capac ty. ence
t e researc project c ana yze t e n uence o
everal tooth flank modifications on the pitting load capacity
of helical gears.
he theoretical part of this research project included
t e eve opment o a new -compat e ca cu at on
met o to eterm ne t e p tt ng oa capac ty o gears,
especially helical gears. This new calculation method analyzes
helical gears more accurately than the existing ISO 6336-2/
DIN 3990–Part 2 and includes the influence of tooth flank
mo cat ons.
es rogram an es ears
One spur and eight helical gears were ground with
different tooth modifications for varying load distribution.
Table 1 summarizes the main geometry and details on tooth
mo cat ons o spur gears an t e e ca gears ,
an t .
he gear type Gk corresponds to the gear type of FVA
284 Ib (Ref. 12) and, except for a small lengthwise crowning
on the pinion, the reference gear of FVA 284 I/II (Ref. 10).
ence t s gear type s use to ver y t e resu ts w t t e spur
gears o an . e e ca gear type as
no modifications, except for a small lengthwise crowning on
the pinion. Type Slk has a long tip relief on the pinion and
wheel to achieve uniform pressure on the line of contact. The
gear type t s mo e w t a generate re e an a s ort
t p re e or opt ma y un orm pressure str ut on. e
necessary modifications were determined according to the
calculation method in Reference 11, which is based on the
pressure distribution calculated with the program RIKOR G
e . . e w o e ca cu at on s nc u e n t e program
RIKOR H e . .
he test gears are made from case-hardened steel8CrNiMo7-6. The gears were case-carburized and ground
after the teeth were cut. The case depth conformed to Eht ≈
0.1 ·n according to DIN 3990–Part .
Measuring Gear Quality
e test gears were measure on a er nge n erg
-coor nate measur ng mac ne. e qua ty was
determined according to DIN 3960 by measuring the profile
along the involute and across the flank, as well as measuring
the pitch deviation and the true running. The quality of all test
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test pinion 1
shaft 1
test wheel 2
shaft 2
slave wheel 4
slave pinion 3 driv ing shaft
load clutch
torque measuring clutch
F gure 2—FZG back-to-back test r g w th a = 2.5 mm.
0%
20%
40%
60%
80%
100%
120%
Gk S0 Slk Stk
f d s f a s e t f g d
� H 1
Figure 3—Experimentally determined endurance contact stress σH1
for
50% failure probability, according to ISO 6336–2.
Table 2:
Main lubricant data of FVA reference lubricantno. 3 + 4 % Anglamol 99
v scos ty
kinematic
v scos ty
at : v mm s
at : v mm s
at : v mm s
ynam c
v scos ty
at : η m as
at 60 °C: η40 m as
at 80 °C: η40 m as
gears averaged a value better than DIN .
test gears a a an roug ness o = . … . µm.
Such a smooth flank roughness minimizes micropitting risk
and its influence on the pitting load capacity.
Test Rig
p tt ng tests were run n ac -to- ac test r gs
w t a center stance = . mm g. . ese test r gs
have a closed-power circuit with helical slave gears. Both test
rigs are equipped with a speed controller in a range of n =0… 3,100 min-1. The helical hand of the slave and test gears
s es gne to cance t e ax a orces on eac s a t.
es on ons
According to the test conditions in Reference 4, the tests
were run with constant load until the pitting area exceeded
4% of the active flank area of one tooth, or until a maximum
running time of 0·10 oa cyc es on t e p n on. test gears
were run n or , oa cyc es on t e p n on, w t a norma
contact stress o H 0
= , mm , an , oa cyc es
on t e p n on w t H 0
= , mm2. e p n on spee was
n1 = 100 min-1 Test lubricant was the FVA-referenced oil
No. 3 mixed with four percent Anglamol 99. Table 2 shows
the lubricant viscosity. The lubricant was injected in the gear
mes w t a temperature o an a vo ume ow o Qe = –
m n. tests were run w t a r v ng p n on an r ven
wheel. The pinion speed was n1 = 3,000 min-1. Consequently,
the circumferential speed at the pitch point was vt = 17.3 m/sfor type Gk and v
t = 16.9 m/s for S0, Slk and Stk.
va ua on o e es esu s
Figure 3 shows the contact stress σ H 1
calculated according
to ISO 6336-2 and based on the fatigue-endurable torque for
0% failure probability. The results are shown in Figure 3.
Figure 4 shows the fatigue-endurable torques on the pinion
o t e our gear types— , , an t —w c ave t e
ame transverse contact ratio of εa = 1. . In comparison to
the referenced spur gear Gk, the best modified helical gear
type Stk transmits approximately 32% higher torque, while
the endurance contact stress according to ISO 6336-2 isower. t out an mo cat ons gear type , t e
increase of torque is about eight percent compared to the spur
gear reference; the endurance contact stress according to ISO
6336-2 is 19% lower.
Summary of test results:
• e ca gears w t equa transverse contact rat o trans er a
higher torque than spur gears.
• For helical gears, the experimentally determined endurance
con ac s ress σ H
is smaller than the value calculated
according to ISO 6336-2.
• oot an mo cat ons t at g ve un orm stressdistribution increase the pitting load capacity.
New DIN/ISO-Compatible Calculation Method
Following is a practical calculation method. This method
is independent of a calculated pressure distribution, and
compat e w t t e actua - – art .
Since the maximum contact stress is approximated with
imple equations, the pitting load capacity can be determined.
It is planned to consider this calculation method in the
ISO 6336-2 as method B.
n , a ca cu at on met o was eve ope ,
based on the knowledge of the local contact stresses. This
method is described in Reference 11.
The modernized calculation method presented here is
called RV II (Rechenverfahren II), and uses double-signed
ar a es e H 0
. e actua - uses no s gne
ariables; e.g., σ H
.
Nominal contact stress σ" H 0
.
See Equation 1 on page 49.
defines the basic value σ" H
used in calculation method II
(RV II):
The factors Z H
, Z E and Z
ε are identical to the same factors
in ISO 6336-2.
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GEAR TECHNOLOGY 00
49
ac orβ. ccor ng to - , Z
β s ca cu ate
w t :
ase on t e resu ts o t e researc projects e .
, e . an t e present researc program, t e
factor Z "β is redefined as
Z "β is an empirical factor, which takes the research results
for helical gears and theoretical thoughts into account. For
gears with only a small helix angle, the factor Z should be
m ar to t e actor Z β, accor ng to - . s actor
is valid for helix angles between 0° and 3 °. The factor Z "β is
hown in Figure .
Contact stress σ" H
. The contact stress σ" H
is calculated
according to Equation 3. Compared to the nominal contactress
H 0, t e contact stress
H cons ers stress ncreas ng
influences resulting from non-uniform load distribution,
additional dynamic force and so on.
he load factors K A, K
V , K
H β and K
H are the same as in the
actual ISO 6336-1.
Factor Z" B/D
. The single-pair tooth contact factor Z " B/D
considers that the relevant contact stress for pitting is not
necessar y t at o t e p tc po nt. t quat on , t e actor Z "
B/D can be calculated to estimate the maximum contact
ress.
ith Z B/D
according to ISO 6336-2 and f Ca1,2
according to
equations ,6,7,8 or 9 (Table 3).
he factor Z B/D
is from ISO 6336-2. The contact stress of
point B (inner point of single contact) is important for spur
gears. o or spur gears t e actor Z B/D
s t e same as t e
factor Z B/D
, according to ISO 6336-2. In this case the factor
f ZCa1,2
is 1.0 (Table 3). Thus the calculation method for spur
gears remains unchanged.
Factor f ZCa
. The tooth flank correction factors f ZCa
1,2
or p n on an w ee cons er t e g er contact stress at
the beginning and the end of the path of contact for helical
gears, for the case where there are inadequate tooth flank
modifications. Table 3 describes the determination of the
tooth flank correction factor.
e toot an correct on actor ZCa
epen s on t e
load and considers inadequate tooth flank modifications
through comparison of the existing and required tooth flank
modifications. Here only the amount of profile modifications is
taken into account, but an adequate length of the modifications
0%
20%
40%
60%
80%
100%
120%
140%
Gk S0 Slk Stk
d s f s d y g f
T 1
F gure 4—Fat gue-endurable torque on p n on of the tested gear types
Gk, S0, Slk and Stk.
0.6
0.7
0.8
0.9
1.0
1.1
1.2
0° 5° 10° 15° 20° 25° 30° 35°
helix angle �
Z �" xv c
ISO 6336-2
RV II, new calculation method
F gure 5—Hel x angle factor "B accord ng to Equat on 2 n compar son to
B accord ng to ISO 6336–2.
Table 3: Determination of the tooth flank
correction factor f ZCa1,2*
spur gears (β = 0 °),
with and without
odifications:ZCa1,2
= 1.0 (5)
elical gears (β > 0 °),
with and without
modifications
(tip relief, root relief):
(6)
1.1≤ f ZCa 1,2
≤ 1.2
Ca: effective amount of profile
odification
Ca0: tooth deflection
elical gears (β > 0 °),with profile and lengthwise
odifications,
calculated with a 3D-load
istribution program for working
load and operated at this load:
ZCa1,2 = 1.0 (7)
elical gears (β > 0 °),
with profile modifications,
esigned for working load and
perated at this load:
ZCa , = 1.1 (8)
elical gears (β > 0 °),
without modifications and with-
out calculating f ZCa
, according
o equation (6):ZCa ,
= 1.2 (9)
u
1u
bd
F Z " Z Z Z " Z Z pσ"
1
t �E H C H 01/2
+
···== �
Equat on .= cos Z
=
cos
1 Z"
H � H v A H 0 B/D1/2 H K K K K " Z "" =
2,1 ZCa B/D B/D f Z Z" =
0
0
ZCa1,2 Ca
CaCa1f
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0%
20%
40%
60%
80%
100%
120%
Gk S0 Slk Stk
d f s f s f s a
ISO 6336-2
RV II, new calculation method
� H 1
� ' ' H 1
200
400
600
800
1000
1200
1400
1600
1800
2000
G S7 S15 S22 S30 S37
d s
f d s f
RV II, new calculation method
ISO 6336-2
T 1 [ N m ]
Table 5: Permissible torque for σHP
= 1500 N/mm2 according to
ISO 6336-2 and to the new calculation method (RV II)
gear
ype ZCa1 Z
B Z
B ´ T
1,ISO6336 [Nm] 1, RVII [Nm]
G 1.0 1.02 1.02 91 91
S7 1.01 1.01 1062 1044
S15 1.01 1.01 182 103
S22 1.0 .0 1372 171
S30 1.0 .0 1675 1256
S37 1.0 .0 1981 1287
s requ re . t e w o e an topo ogy s to e cons ere , t e
calculation method according to Reference 11 can be used.
For spur gears, the tooth flank correction factor f ZCa
1,2
is
.0. Hence it is assumed that the contact stress of the inner
point of single contact (B) is higher than the contact stress of
t e start ng po nt w t ou e-toot contact, w c s t e
usual case for practical spur gears.
If helical gears have no tooth modifications or only
tandard modifications—like tip or root relief—the factorf ZCa1,2
is calculated according to Equation 6.
e va ue ranges rom . to . . or e ca gears
with adequate tooth modifications, which cause maximum
contact stress near the pitch circle, the factor f ZCa
1,2
is 1.0.
These adequate tooth modifications are normally generated.
For helical gears with tooth modifications designed for the
wor ng oa an operate at wor ng o a , t e actor ZCa
1,2
is 1.1. For helical gears without modifications a tooth flank
correction factor f ZCa
1,2
= 1.2 can be estimated.
Existing and required profile modifications. The amount
of profile modification C a in Equation 6 is the actual profilemo cat on on t e toot an n m crons. C
a0 s t e requ re
amount of profile modification in microns for having the
maximum contact stress around the pitch point, not at the
beginning or the end of the path of contact. The required
profile modification can be calculated according to the
manu acturer s exper ence or recommen at ons, accor ng to
Niemann/Winter (Ref. 6), Sigg (Ref. 9) or Equation 10.
(10)
e actor C a0 epen s on t e oa or a recommen at ons.According to Equation 10, the required profile modification is
calculated by the nominal tangential load F t , the face width
and the stiffness c γ , according to ISO 6336-1.
Applying the New DIN/ISO-
ompa e a cu a on e o
The following contact stresses are calculated from the
fatigue-endurable torque. Figure 6 shows the contact stresses
on pinion σ H 1
of the test gears according to ISO 6336-2, as
well as the contact stress σ H 1
, as determined by the new
-compat e ca cu at on met o . v ous y t e
calculation method RV II gives the same contact stress for the
pur gear as ISO 6336-2. However, for helical gears a higher
contact stress is calculated.
The contact stresses of the different gear types σ H 1
accor ng to - range n a sprea o ± o t e
mean value. According to RV II, the contact stresses σ" H 1
range in a spread of ± 2% of the mean value. Especially for
the helical gears with practice-relevant modifications (Slk
and Stk), the calculated contact stresses (RV II) are nearly the
ame as t e contact stress o t e spur gear.
Consequently the new calculation method RV II
corresponds better to test results than the ISO 6336-2
calculation method.
F gure 6—Contact stress accord ng to ISO 6336–2 and to the new
calculat on method RV II.
Figure 7—Permissible torque according to ISO 6336–2 and to the new
calculat on method RV II.
=c
b / F Ca
t 0
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Theoretical Study to Determine the
In uence of Helix Angle on the Endured Torque
e o ow ng stu y est mates t e perm ss e torque as a
function of the helix angle according to the new calculation
method. Therefore the contact stresses according to RV II for
ix different gears (Table 4) were calculated. All gears have
t e same center stance, ace w t , transverse pressure
ang e, transverse mo u e, num er o teet , transverse contact
ratio and constant tip diameter. The permissible contact stressis the same as in ISO 6336-2.
he maximum endurable torque can be determined for
t e con t on w ere t e contact stress accor ng to t e new
ca cu at on met o s equa to t e perm ss e contact stress
according to ISO 6336-2:
perm ss e contact stress o HP
= , mm s
assumed for all gears. The load factors and f ZCa
1 are set to 1.0
(K ges = K A · K v· H β· K H = 1.0 and f ZCa 1 = 1.0).
onsequent y, t s assume t at a e ca gears ave
un orm oa str ut on, resu t ng rom a equate an
modifications.
he calculated torque T 1 on the pinion and the main factors
according to the new calculation method are printed in Table ,
an t e perm ss e torques are s own n gure . v ous y,
t e perm ss e torque ncreases w t ncreas ng e x ang e.
In comparison to the spur gear (G), the helical gear S30, with
a helix angle of 30˚, has a 30% higher calculated torque.
Conclusionse resu ts o t e exper menta part o t s researc project
prove that the calculation of the contact stress according to
ISO 6336-2/DIN 3990–Part 2 isn’t accurate for helical gears.
ence a new calculation method to determine the pitting
load capacity was developed. This calculation method is
compat e w t - – art . t ta es t e
maximum contact stress into account and can be used as a
new method B. This calculation method, called RV II, treats
helical gears more precisely than ISO 6336-2, and is capable
of accounting for tooth modifications. For spur gears the new
ca cu at on met o s equa to - .
A knowledgments
The project “Einfluss der örtlichen Hertz´schen
Pressung auf die Grübchentragfähigkeit einsatzgehärteter
Stirnräder” was unded by the FVA—Forschungsvereinigung
Antriebstechnik e.V. (FVA 284 Ic).
References
. DIN 3990, Teil 2: Tragfähigkeitsberechnung von Stirnrädern
- erec nung er r c entrag g e t. eut er ag er n,
987.
2. Döbereiner, R. Tragfähigkeit von Hochverzahnungen Gerin-
ger Schwingungsanregung. FVA-Vorhaben 2 7, Heft 71,
51
ran urt, .
3. Haslinger, K. Untersuchungen zur Grübchentrag fähigkeit
Profilkorrigierter Zahnräder . Dissertation TU München,
991.
4. Hösel T. and J. Goebbelet. Empfehlungen zur Vereinheit-
c ung von an entrag g e tsversuc en an er teten
und Gehärteten Zylinderrädern. FVA-Merkblatt Nr. 0/ ,
Frankfurt, 1979.
. ISO 6336-2: “Calculation of Load Capacity of Spur andHelical gears—Calculation of Surface Durability (Pitting).”
rst e t on, .
6. Niemann, G. and H. Winter. Maschinenelemente Band II .
Springer Verlag Berlin, Heidelberg, New York, London, Paris
and Tokyo, 1989.
7. Otto, M. “Ritzelkorrekturprogramm RIKOR H.” FVA-
orsc ungsvor a en , e t , .
8. Schinagl, S. “Ritzelkorrekturprogramm RIKOR G.” FVA-
Forschungsvorhaben 30/IV, Heft 481, 2000.
. Sigg, H. Profile and Longitudinal Corrections on Involute
Gears. Semi Annual Meeting of the AGMA, Paper 109.16,.
0. Stahl, K. and O. Hurasky-Schönwerth. Experimentelle
und Theoretische Untersuchungen an Schrägstirnrädern.
FVA-Vorhaben 284 I/II, Heft 608, Frankfurt, 2000.
1. Stahl, K. Vorschlag für eine Erweiterte Berechnungs-
met o e zur r c entrag g e t nsatzge rteter
gerad- und Schrägverzahnter Stirnräder. FZG-Bericht 2 0,
München, 2000.
2. Steinberger, G. DIN-Vergleichstest zur Grübchen-
tragfähigkeit gerad- und Schrägverzahnter Stirnräder. FVA-
or a en , e t , ran urt, .
HP H " =